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{"type":"item.completed","item":{"id":"item_2","type":"agent_message","text":"```python\nimport numpy as np\nfrom scipy.linalg import svd, eigvals\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(limits=2)\n\nd=384\nP=np.eye(d)-np.ones((d,d))/d\nrs=[]\nfor h in [1,5]:\n    _,_,r=svd(np.asarray(W[\"W_v.0\"][64*h:64*(h+1)],float)@P,full_matrices=False)\n    rs.append(r[:28])\nee,U=np.linalg.eigh(sum(r.T@r for r in rs))\nU=U[:,np.argsort(ee)[::-1][:28]]\nR=P-U@U.T\nq5=np.asarray(W[\"W_q.0\"][320:384],float)@R\nk5=np.asarray(W[\"W_k.0\"][320:384],float)@R\ngg=(q5.T@q5+k5.T@k5)/2\nee,V=np.linalg.eigh(gg)\nix=np.argsort(ee)[::-1]\nee,V=ee[ix],V[:,ix]\nT=V[:,:24]/np.sqrt(ee[:24])[None,:]\na5=T.T@(q5.T@k5)@T\nleft,_,right=svd(a5)\nrot=left@right\npowers=list(range(-3,9))\n\nprint(\"L0 position-subspace bilinear forms versus powers of the H5 polar factor\")\nprint(\"Power columns:\",powers)\nfor h in range(6):\n    s=slice(64*h,64*(h+1))\n    q,k=[np.asarray(W[f\"W_{n}.0\"][s],float)@R for n in [\"q\",\"k\"]]\n    a=T.T@(q.T@k)@T\n    sims=[]\n    for p in powers:\n        rp=np.linalg.matrix_power(rot,p)\n        sims.append(np.sum(a*rp)/(np.linalg.norm(a)*np.linalg.norm(rp)))\n    print(f\"L0H{h}\",np.round(sims,3))\n\nA={};B={}\nfor l in range(4):\n    scale=np.asarray(W[f\"ln1_w.{l}\"],float)\n    for h in range(6):\n        s=slice(64*h,64*(h+1))\n        q,k,v=[np.asarray(W[f\"W_{n}.{l}\"][s],float) for n in [\"q\",\"k\",\"v\"]]\n        A[l,h]=P@(scale[:,None]*(q.T@k)*scale[None,:])@P\n        B[l,h]=np.asarray(W[f\"W_o.{l}\"][:,s],float)@(v*scale[None,:])@P\n\nprint(\"\\nMatching geometry restricted to initial shared value-reading subspace\")\nprint(\"For U.T QK_l OV_0H5 U: trace/Frobenius, trace/nuclear, symmetric fraction, largest eigenvalues\")\nfor l in range(1,4):\n    for h in range(6):\n        c=U.T@A[l,h]@B[0,5]@U\n        ss=np.linalg.svd(c,compute_uv=False)\n        ev=eigvals(c)\n        ev=ev[np.argsort(np.abs(ev))[::-1]][:4]\n        print(f\"L{l}H{h}\",\n              round(np.trace(c)/np.linalg.norm(c),3),\n              round(np.trace(c)/ss.sum(),3),\n              round(np.linalg.norm((c+c.T)/2)**2/np.linalg.norm(c)**2,3),\n              np.round(ev,3))\n\nprint(\"\\nQK affinity of L1 heads: pairwise Frobenius cosines\")\nprint(np.round([[np.sum(A[1,h]*A[1,k])/\n                 (np.linalg.norm(A[1,h])*np.linalg.norm(A[1,k]))\n                 for k in range(6)] for h in range(6)],3))\n\nprint(\"\\nDirect positional-weight overlap at later layers\")\n# This tests only an unchanged residual-coordinate subspace, not layer outputs.\nfor l in range(1,4):\n    for h in range(6):\n        c=T.T@A[l,h]@T\n        sims=[]\n        for p in powers:\n            rp=np.linalg.matrix_power(rot,p)\n            sims.append(np.sum(c*rp)/(np.linalg.norm(c)*np.linalg.norm(rp)))\n        k=int(np.argmax(sims))\n        print(f\"L{l}H{h}\",\"best polar power\",powers[k],\n              \"cosine\",round(sims[k],3),\n              \"at powers 0,1,2\",np.round(sims[3:6],3))\n```"}}
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